Analysis of Hydrologic Drought Frequency Using Multivariate Copulas in Shaying River Basin

Abstract

Droughts, considered one of the most dangerous and costly water cycle expressions, always occurs over a certain region, lasting several weeks or months, and involving multiple variables. In this work, a multivariate approach was used for the statistical characterization of hydrological droughts in Shaying River Basin with data from 1959–2008. The standard runoff index (SRI) and the run theory were employed to defined hydrological drought character variables (duration, severity, and intensity peak). Then, a multivariate joint probability analysis with four symmetric and corresponding asymmetric Archimedean Copulas was presented; and the multivariate frequency analysis with the joint return periods ($T_{and}$ and $T_{or}$) were estimated. The results showed that the hydrological droughts have a severity of 4.79 and 5.09, and the drought intensity peak is of 1.35 and 1.50 in Zhoukou station and Luohe station, respectively; the rank correlation coefficients τ are more than 0.5, which means multivariate copulas can effectively describe the joint frequency distributions among multivariate variables. Drought risk shows a spatial variation: the downstream observed at Zhoukou station is characterized by a higher multivariate drought risk. In general, multivariate copulas provide a reliable method when constructing a comprehensive drought index and evaluating multivariate drought characteristics. Thus, this paper can provide useful indications for the multi-dimensional droughts’ risks assessment in Shaying River Basin.

1. Introduction

A drought is generally considered a natural phenomenon that occurs when rainfall and runoff are significantly lower than the normal value. A typical drought always brings undesirable damages to the development of regional economy and agriculture caused by precipitation and water supply shortage [1,2]. According to statistics, global economic losses due to droughts reached up to 6–8 billion dollars, far exceeding losses caused by other meteorological disasters [3]. Therefore, it has been a challenging topic in water resources management. Therefore, an accurate drought risk assessment is fundamental to prevent and mitigate drought disasters [4].

Hydrologic drought caused by streamflow deficits is one important type of drought event. The quantitative identification of a hydrologic drought event is always an important task and the basis of the drought risk assessment and mechanism study. Operational definitions, which consider main drought features such as onset, termination, duration, severity, and intensity, are necessary to quantitatively address the problem [5]. A drought event is a multivariate phenomenon [6]. A criticism to traditional approaches is that hydrological droughts are described only through one drought characteristic, the maximum water deficit, without considering the length of the drought and the total water deficit [7], which may not reflect the complex characteristics of drought events [6]. Therefore, one purpose of this study is the multivariate drought risk assessment considering multiple drought characteristics.

Yevjevich (1967) proposed another approach for drought analysis, which applied runs theory for drought events identification [8]. According to this theory, a threshold level is selected to determine drought events, and with the threshold, the drought is characterized by the duration, the interval between two drought events, and the severity. The runs theory has been widely applied, and this rationale has been followed by many hydrologic drought analyses [9,10,11]. However, the determination of the threshold demand level relies on subjective assumptions, and it is possible that a small change of the threshold will dramatically alter the abstracted events. Furthermore, the application of the run method also requires particular attention to the possible presence of mutually “dependent” droughts. That is to say, the fact that a prolonged dry period may be interrupted by shorter ones where the flow exceeds the threshold level, and a long drought turns out to be divided into a number of shorter ones. In some research, the threshold has been fixed as equal to the long-term mean annual flow, or the percentage of 75% or 90% of the mean annual flow [12,13]. In this paper, the Standardized Runoff Index (SRI) [14] was used to describe the runoff droughts with the threshold determined based on the SRI classified standards, and the drought characters defined with the run theory.

As the drought characteristics are random variables, they are always evaluated and modelled with probability theories when they were defined. In literature, until 2000, the drought frequency analysis was addressed principally under a univariate framework by calculating the probability distribution of drought duration and drought severity and considering these variables as independent. Droughts are multiplex phenomena and drought characteristics are also mutually correlated variables [6,15]; univariate analysis cannot give an exhaustive appraisal of droughts and account for significant correlations among their characteristics [16]. Thus, a multivariate probabilistic framework is advisable for a proper description of droughts. Therefore, many efforts have been devoted to examining drought risk on multivariate probability levels to give a more outright drought characterization [17,18]. However, the traditional multivariate distributions generally have various constraints. For instance, the marginal distribution of the drought characteristics must come from the same distribution type [19]. The copula function overcomes major drawbacks inherent in the traditional methods [15,20], and has been extensively employed in multivariate analysis. A copula is a function that compounds various univariate cumulative distribution functions (CDF) into their joint CDF [21,22]. The main advantages of copulas are that they are capable of developing the multivariate distribution in simple terms regardless of the univariate distribution type and preserving the dependence structure among variables. Hence, copulas have gained wide popularity in multivariate drought modeling and successful results have been obtained in real case studies [16,23,24,25,26].

For analysis of drought variables, the joint distribution using copulas provides an objective description of the overall deficit status and paves the path for the better estimation of probabilistic quantities such as return period, persistence at a given severity, associated risk assessment, and improved identification of possible portents of droughts. Many studies have reported the multivariate modeling of droughts using copula functions [15,27,28,29,30]. Most of the above-mentioned works focused on hydrologic droughts with bivariate copula of duration and severity, and some attention was paid on multivariate copula with three hydrologic drought characters [31,32,33]. Song and Singh (2010b) addressed hydrological droughts via a trivariate analysis of duration, severity, and inter-arrival time, as identified by means of a threshold on the flow discharge [34]. Furthermore, Song and Kang (2011) considered the pair-copula construction method to build high-dimensional dependence structures, and carried out a trivariate analysis of the variables: duration, severity, and severity peak, as identified by the means of a threshold on the flow discharge [35]. Results of these studies demonstrate that trivariate copulas are robust for hydrologic drought analysis.

In summary, we introduced the application of SRI, copulas, and trivariate copulas in hydrological drought. However, few studies combine the three into one study. In this paper, we used the SRI series and the run theory to compute drought duration ($D$), severity ($S$), and peak ($P$) of 1951 and 2008. In addition, we identified marginal distribution for drought duration, severity, and intensity peak series, and identified the most suitable copula functions for joint frequency distribution [36,37,38,39,40]. Moreover, we appraised the performance of four bivariate and eight trivariate Archimedean copulas and computed the probabilistic drought properties using the best-fitted copula functions. Results of this study will further provide a framework for regarding climate trends and climate change future projection on seasonal time step in SYRB [39,40].

2. Study Area and Data

The Shaying River, the longest tributary of the Huai River, originates from the FuNiu Mountain in Henan Province, and flow into the Huai River in Anhui Province with the stream length 624 km. The Shaying River Basin (SYRB), the portion of the Huai River Basin, is located in North China (32°20′–34°34′ N and 112°45′–113°15′ E), whose area is 39,880 km² (Figure 1).

The climate in the SYRB is characterized by arid and semiarid continental monsoon. The average annual precipitation is approximately 950 mm, and almost 60% occurs throughout the monsoon season. The precipitation has a larger interannual variation with a maximum annual precipitation of 1616.3 mm and a minimum 447.1 mm. The average annual runoff mainly sourced from precipitation in the SYRB is 145.4 mm and almost 70% of the annual runoff is concentrated in the flood season. Due to the inter-annual and seasonal changes in precipitation and runoff, drought and flood disasters occur frequently in the SYRB [41,42].

In the SYRB, agriculture is the primary part of the basin economy, and most of the habitants’ income was directly or indirectly related to agriculture. Irrigation is used on the cultivated land, which is anticipated to increase in the future because of the projected change in climate and the gradual decline in groundwater over the years. This makes the agriculture sector highly susceptible to drought, which is a recurrent phenomenon in most parts of the basin. Better characterization of drought and associated risks is vital for executing mitigation plans to lessen drought impact, particularly related to observed and anticipated climate changes in the region [43].

In the study, the monthly precipitation data from 1951 to 2008 were obtained from 17 hydrological stations in the basin (Figure 2). Monthly runoff data from 1951 to 2008 were acquired from two hydrology stations, Luohe station in Sha River, Huangqiao station in Ying River, Fugou station in Jialu River, and Zhoukou station in Shaying River. In addition, the temperature data were obtained from five selected meteorology stations in the basin from China Meteorological Data Network (http://data.cma.cn/ accessed on 1 March 2022).

3. Methodology

3.1. Construction of the Copula-Based Joint Drought Deficit Index

3.1.1. Standardized Runoff Index (SRI)

In this study, the standardized runoff index (SRI) [44] was adopted for statistical analyses of hydrologic variables. It is a probability measure of cumulative runoff by definition. Therefore, drought severity is scaled in terms of probabilities that can be compared between various locations and among variables. The SRI was calculated by fitting a cumulative probability density function (CDF) with the runoff records and then transformed into a standard normal distribution for the actual SRI values using an equal probability transformation. Different time scales of SRI values can be calculated from one to dozens of months (e.g., a 3-month scale represents seasonal droughts and is related to soil moisture conditions) [45,46]. The SRI classification of hydrological drought severity described in GB/T 20481-2017 (as SPI) is shown in

Table 1. Drought classification according to SPI index.

Degree	Classification	SRI
1	No drought	−0.5 < SRI
2	Mild drought	−1.0 < SRI ≤ −0.5
3	Moderate drought	−1.5 < SRI ≤ −1.0
4	Severe drought	−2.0 < SRI ≤ −1.5
5	Extreme drought	SRI ≤ −2.0

.

3.1.2. Drought Characters Identification with Runs Theory

Run theory is an approach used for the identification of drought characters including drought duration and severity [47]. Based on a given demand (for instance, mean discharge), the observed time series are divided into wet events (values greater than demand) and dry events (values less than demand). If drought events over a certain period have a drought index value that remains less than demand, the run is considered a negative run [48].

The characters of the drought events were defined based on run theory as shown in Figure 3. The minimum flow in a drought event is of great significance for water resource management, for example affecting irrigation, hydropower, and shipping. The minimum flow in a drought event which is defined as the drought intensity peak ($P$) in a drought period is also considered as a vital character. Therefore, this paper analyzed the hydrological drought characters, duration, severity, and intensity peak. In Figure 3, $\mathrm{duration} \left(D\right)$ was defined as drought duration which means the period during drought appeared with the SRI value continuously below the threshold. $\mathrm{Severity} \left(S\right)$ was defined as drought severity, the cumulative values of SRI for the drought duration, namely the shadow area. P was defined as the drought intensity peak, which is the absolute value of the minimum SRI in a drought event, namely the extreme value of the negative run.

The steps of the definition of drought characters were as follows:

(1)

According to the SRI drought degree classification presented in Table 1, three given threshold degree of runs theory were defined ($SRI=-1,-0.5, \mathrm{and} 0$). When the monthly SRI values were below $-0.5$, the corresponding month was potentially identified as a drought event (e.g., the four drought events shown in Figure 3, including a, b, d, e, f).

(2)

For a drought event whose duration was only one month (e.g., a and f), if SRI < −1, this month was regarded as a single drought event (e.g., with a severity $Sa$), or else non-drought event in the month (e.g., f)

(3)

A drought event may contain a few consecutive months with negative SRI (e.g., b and its severity $Sb$).

(4)

If the duration of a drought event contained two branches, (e.g., d and e), and c is the interval between drought event d and e, and the duration of c was less than 6 months, in which $-0.5<SRI<0$, then drought d and e were still regarded as a single drought event, drought duration $D=De+Dd+Dc$, the corresponding severity (shadow area in Figure 3) was defined as $S=Se+Sd$. Otherwise, e and d were recognized as two independent drought events.

(5)

P was the absolute value of minimum SRI value in a drought event (e.g., $Pb, Pd$).

3.2. Copula-Based Joint Distribution Function

The copula-based multivariate distribution function may couple marginal distributions into bivariate or multivariate distribution [49]. Accordingly, copulas are widely used in the joint probability analysis in such area as finance, hydrology, meteorology, and risk management [21,49].

As Tarski’s theorem [50], $\left(x_1,x_1,\cdots x_n \right)$ were n-dimensional continuous random variables and $F_1\left(x_1\right),F_2\left(x_2\right),\cdots ,F_n\left(x_n\right)$ were marginal cumulative distribution functions (CDFs), there exists one unique n-copula function C:

$$F\left(x_1,x_1,\cdots x_n \right)=P\left\{X_1\le x_1,X_2\le x_2,\cdots X_n\le x_n\right\}=C\left[F_1\left(x_1\right),F_2\left(x_2\right),\cdots ,F_n\left(x_n\right)\right]$$

(1)

Copulas have been constituted from different families and the following types were used to simulate the joint distribution of trivariate drought variables in this study: four kinds of symmetric Archimedean Copula functions (Frank, Clayton, Gumbel, and Joe) and four asymmetric Archimedean Copula functions ($M3, M4, M5, \mathrm{and} M12$). Furthermore, the parameters θ of copula functions were estimated using the semi-parametric maximum estimation method [51,52]. The Root Mean Square Error (RMSE) criterion and Akaike Information Criterion (AIC) were used to test the goodness of simulation. The smaller the RMSE and AIC, the better the fitting goodness of the parameter is, and the minimum RMSE and AIC means the optimal fitting result. RMSE and AIC are as follows:

$$\{\begin{array}{c}MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left(p_{ei}-p_i\right)}^2\\ AIC=nln\left(MSE\right)+2k\\ RMSE=\sqrt{MSE}\end{array}$$

(2)

where n is the sample capacity; $k$ is the number of copula parameters; $i$ is the sample serial number; $P_i$ is the value of joint distribution of the $i\mathrm{th}$ copula function; $P_{ei}$ is the $i\mathrm{th}$ empirical value; and MSE is the mean square error.

3.3. Marginal Distribution

When copulas were used to analyze the droughts characters’ joint probability functions, it was a key step to identify and determine the most appropriate marginal distribution for each of the dependent variables. In this study, four distributions of exponential, gamma, log-normal, and Weibull were directly examined to determine the most appropriate marginal probability distribution for drought duration, severity values, and intensity peak. Then, the Maximum Likelihood Estimation (MLE) was used to estimate the shape, scale, and position parameters of the four marginal probability distributions [50]. The Kolmogorov–Smirnov (K–S) test statistic [53] was used to fit the goodness and further select the optimal distribution function. The Kendall rank correlation coefficient was used to test the degree of independence among the variables [54,55].

3.4. Return Period Estimation

The joint drought frequency was analyzed in terms of the joint return periods, which were the average elapsed time between the occurrences of two droughts with specific characteristic variables. According to the theory of return period [56], the return period of univariate variable can be estimated by:

$$T=\frac{N}{m\left(1-F\left(x\right)\right)}$$

(3)

where $F\left(x\right)$ is the distribution function of drought characteristic variables, $N$ is the length of the data, and $m$ is the number of the drought events.

However, a drought is considered as a multivariate event characterized by drought duration, severity, and intensity peak, the joint return period estimated by bivariate and trivariate drought characteristic variables provides more useful information for drought assessments. Therefore, in this study, the joint return periods using the methodology proposed were estimated. The joint return periods were calculated for two cases; the return period Tand for $D\ge d$ and $S\ge s$ (AND case) and the return period Tor for $D\ge d$ or $S\ge s$ (OR case). The frequency distribution function of drought duration $D$, severity $S$, and intensity peak $P$ are assumed as $u$, $v$, and $S$, respectively, then $T_{and}$ and $T_{or}$ between drought duration $D$ and severity $S$ are estimated:

$$T_{and}=\frac{N}{nP\left(D\ge d\cup S\ge s\right)}=\frac{N}{n\left[1-C\left(u,v\right)\right]}$$

(4)

$$T_{or}=\frac{N}{nP\left(D\ge d\cap S\ge s\right)}=\frac{N}{n\left[1-u-v+C\left(u,v\right)\right]}$$

(5)

where $d$ and $s$ are the given drought duration and severity, respectively.

The $T_{and}$ and $T_{or}$ of drought duration $D$, severity $S$, and intensity peak $P$ are as follows, respectively:

$$T_{and}=\frac{N}{nP\left(D\ge d\cup S\ge sW\ge w\right)}=\frac{N}{n\left[1-C\left(u,v,w\right)\right]}$$

(6)

$$\begin{gathered} T_{or}=\frac{N}{nP\left(D\ge d\cap S\ge s\cap W\ge w\right)} \\ =\frac{N}{n\left[1-u-v+C\left(u,v\right)+C\left(u,w\right)+C\left(v,w\right)-C\left(u,v,w\right)\right]} \end{gathered}$$

(7)

4. Results and Discussion

4.1. Drought Characteristics Identification and Analysis

With the monthly runoff data (1951–2008) observed from the Luohe and Zhoukou hydrological station, drought events were identified by SRI and run theory. The SRI-3 drought index, which represents the 3-month runoff anomalies, is used to represent the drought time series. The results show that the basin experienced major droughts during 1966, 1978, 1979, 1997, and 2000, and the statistical results of drought characteristic variables were shown in

Table 2. Hydrological drought characteristics ($D, S$ and $P$).

Period	Luohe Station				Zhoukou Station
	Times	D (Months)	S	P	Times	D (Months)	S	P
1950s	5	4.75	5.33	1.29	2	2.00	2.90	1.61
1960s	9	4.43	5.24	1.52	8	3.13	3.46	1.33
1970s	13	3.92	3.72	1.19	10	3.10	5.18	1.66
1980s	7	5.29	4.48	1.40	8	3.88	4.07	1.36
1990s	8	5.88	6.05	1.24	8	7.25	8.08	1.52
2000s	9	4.88	4.55	1.44	5	4.00	5.12	1.53
1951–2008	51	4.78	4.79	1.35	41	4.17	5.09	1.50
Maximum duration	14 months				19 months

.

In total, there were 51 drought events that occurred over the 50-year period in the upstream observed by the Luohe station. The average duration ($D$) was 4.78 months, average drought severity ($S$) was 4.79, and the average intensity peak $P$ was 1.35. The longest drought duration was 14 months. During the past 50 years, the drought events occurred more but with shorter duration and lower severity in the 1970s, while the drought in the 1990s was the heaviest with the longest drought duration and the highest drought severity.

There were 41 drought events in the downstream observed by Zhoukou station, and the average drought duration ($D$) was 4.17 months, the average severity ($S$) was 5.09, and the average intensity peak ($P$) was 1.5. The longest drought duration was 19 months. Temporal distribution in the downstream of mean drought duration, severity, and intensity peak showed a different pattern. The downstream in the 1970s had more drought events and higher severity but with shorter duration, which showed that the drought was relatively severe in the 1970s, and the drought events in the 1990s occurred with the longest duration, and the highest severity, which means that the heaviest drought was experienced for the downstream in the 1990s.

Spatial distribution of drought duration, severity, and intensity peak showed that the drought events in the upstream had more frequency (51 times) with a longer duration (4.78 months) but lower severity and peak (4.79 and 1.35) than that in the downstream (times: 41; duration: 4.17; severity: 5.09; peak: 1.50). It means that drought deficit in the downstream was more serious than that in the upstream.

4.2. Univariate Marginal Distribution Selection

Here, the univariate marginal distribution of gamma, exponential, log-normal, and Weibull distribution were used to fit the drought duration, severity, and intensity peak, and the parameters of the distributions were estimated by the maximum likelihood method and are presented in

Table 3. Univariate marginal distribution and parameters estimation.

Hydrological Station	Drought Characteristic Variable	Marginal Distribution Function	Parameters
			Shape Parameter	Scale Parameter	Location Parameter
Luohe	D	Exponential	-	3.83	0.43
	S	Log-normal	0.85	-	1.08
	P	Gamma	10.14	0.13	-
Zhoukou	D	Gamma	1.64	2.55	-
	S	Weibull	1.08	5.26	-
	P	Log-normal	0.39	-	0.32

. The K–S test was used to choose the fit goodness of the marginal distribution. The marginal distribution function that has the minimum statistical value ($Z$) and higher confidence level ($\beta$) was considered as the best-fit one. The goodness-of-fit results are shown in

Table 4. K–S Fitting test of marginal distribution function and correlation coefficients for D–S, D–P, and S–P.

Hydrological Station	Drought Characteristic Variable	K–S Result			Correlation Coefficient τ
		Z	β	Critical Value Z (α = 0.05)	D–S	D–P	S–P
Luohe	D	0.1625	0.14	0.191	0.8334	0.5241	0.5971
	S	0.1532	0.17
	P	0.0537	0.99
Zhoukou	D	0.1409	0.36	0.2076	0.8537	0.5300	0.66
	S	0.1556	0.26
	P	0.0819	0.93

. The statistical value ($Z$) of $D$, $S$, and $P$ marginal distributions were all smaller than the critical values ($Z\alpha =0.05$), and $\beta$ values were greater than α = 0.05. This also means the marginal distribution functions fit well with drought characteristic variables ($D$, $S$, and $P$).

As shown in Table 3, the exponential distribution fits better for the drought duration in upstream, while the gamma distribution is better in downstream. Log-normal distribution fits better for the drought severity in upstream, while the Weibull is better in the downstream; and gamma distribution and log-normal fit better for the intensity peak in upstream and downstream, respectively.

The dependence among drought severity, duration, and peak was tested by the nonparametric Kendall rank correlation coefficient ($\tau$). All the rank correlation coefficients τ between $S–D, S–P$ and $D–P$ were more than 0.5, indicating that a strong positive correlation between any two different drought characteristic variables and the copula function could be used to construct the joint distribution. Because of the dependency, the joint probability of drought variables could be modeled with bivariate or multivariate probability distribution instead of the marginal distribution.

4.3. Bivariate and Trivariate Copula Joint Distribution Model of Drought

4.3.1. Bivariate Copula Model

Four widely used copula functions, including Frank, Clayton, Gumbel, and Joe, were used to fit for the bivariate joint distribution model of hydrological drought characteristic variables. Copula parameters were estimated by semi-parametric estimation method. The RMSE and AIC criterion were used to test the goodness of fit and select the best-fit copulas. The results of parameters, RMSE, and AIC are presented in

Table 5. Parameter, RMSE, and AIC of bivariate drought variables.

Copula Function	Variable	Luohe			Zhoukou
		θ	RMSE	AIC	θ	RMSE	AIC
Frank	D–S	17.6	0.0768	−259.75	19.82	0.0822	−202.92
	D–P	2.97	0.0681	−272.06	5.84	0.0687	−217.55
	S–P	5.08	0.0791	−256.81	8.28	0.0555	−235.03
Clayton	D–S	3.28	0.0932	−240	4.25	0.0956	−190.52
	D–P	0.59	0.078	−258.24	1.2	0.0839	−201.19
	S–P	1.03	0.0428	−319.38	1.25	0.0842	−200.95
Gumbel	D–S	4.39	0.0842	−250.44	4.85	0.0892	−196.18
	D–P	1.5	0.0664	−274.63	1.91	0.0761	−209.22
	S–P	1.9	0.0475	−308.73	2.38	0.062	−225.99
Joe	D–S	5.98	0.0909	−242.59	6.37	0.0992	−187.47
	D–P	1.81	0.0674	−273.12	2.26	0.0856	−199.54
	S–P	2.34	0.0545	−294.81	3.1	0.0789	−206.24

.

As the results show in Table 5, in Luohe station, Frank copula, which has minimum RMSE and AIC, is considered as the best-fit bivariate joint distribution of drought duration and severity; Gumbel copula was the best-fit distribution for drought duration and intensity peak; and Clayton copula fit best for drought severity and intensity peak. In Zhoukou station, Frank copula was found to fit best for any two drought characteristic variables. Taking Zhoukou station as example, the Q-Q plots (Figure 4) closing to the 45° diagonal line also support the choice of Frank copula.

Taking the results observed in Zhoukou station as an example, the bivariate Frank copula joint cumulative probability distribution and the contour lines of $S–D, S–P$ and $D–P$ are depicted in Figure 5.

Given a drought characteristic variable ($D, S,$ or $P$), the joint probability increased with the other variable’s increase, and up to a certain value. As shown in Figure 5a, supposing $D=6$, the joint probability increased from 0.1 to 0.7 with the S’s increase, supposing $S=5$, the joint probability rose from 0.1 to 0.6 with the D’s increase; with $D$ and $S$ both growing indefinitely, the joint probability increased up to 1. The similar laws of $D–P$ and $P–S$ were displayed in Figure 5b,c.

The joint probability contours of $S–D, S–P, \mathrm{and}$ $D–P$ are shown in Figure 5d–f. The amplitude of contour depression of joint probability distribution was the largest, that is, under the condition of given joint probability value, only a small edge probability could meet the joint distribution, indicating that the correlation between drought duration and drought intensity was the highest. It can be seen that the amplitude of isoline depression was the smallest, and the correlation between drought duration and drought intensity was the lowest.

The joint probability contour lines help to determine the joint return periods ($T_{and}$ and $T_{or}$) if the mean values of drought duration, severity, and intensity peak are known.

4.3.2. Trivariate Copula Function Model

Four symmetric Archimedean copula functions (Frank, Clayton, Gumbel, and Joe) with one parameter and the corresponding asymmetric Archimedean copula functions ($M3, M4, M5, M12$) with two parameters were used to fit the trivariate joint distribution of drought duration, severity, and intensity peak. The copula parameters ($\theta \mathrm{and} \theta 1, \theta 2$) were estimated by semi-parametric estimation method, and then the goodness of fit was evaluated with the RMSE and AIC citation. The results are presented in

Table 6. Parameters, RMSE, and AIC of trivariate drought variables.

	Symmetric Archimedean Copula				Asymmetric Archimedean Copula
Hydrological Station	Copula Function	θ	RMSE	AIC	Copula Function	θ1	θ2	RMSE	AIC
Luohe	Frank	5.38	0.0851	−249.39	M3	3.32	5.68	0.0971	−235.85
	Clayton	1.15	0.1061	−226.81	M4	1.01	1.98	0.0879	−245.92
	Gumbel	1.94	0.0916	−241.83	M5	1.51	1.99	0.0999	−232.99
	Joe	2.39	0.1058	−227.14	M12	1.91	2.83	0.1241	−210.81
Zhoukou	Frank	8.38	0.0819	−203.13	M3	6.31	9.84	0.0673	−219.23
	Clayton	1.48	0.1223	−169.86	M4	2.26	3.91	0.0834	−201.66
	Gumbel	2.39	0.1772	−139.89	M5	2.13	2.95	0.0704	−215.54
	Joe	2.97	0.1146	−175.64	M12	3.11	4.71	0.1027	−184.66

.

In Luohe station, Frank copula, with the smallest RMSE and AIC, was considered the best-fitting trivariate joint distribution function of drought duration ($D$), severity ($S$), and intensity peak ($P$), and M3 copula in Zhoukou station fitted best of $D, S,$ and $P$. The Q-Q plots (Figure 6), near to the 45° diagonal line, also support the results of the best-fitting copulas, especially since the upper tail is sharper in the Q-Q plot of the M3 copula, which is of vital significance to improve the accuracy of drought risk assessment [57,58]. Therefore, Frank copula and M3 copula could be used as conjunction functions to simulate the joint distribution of trivariate drought characteristic variables.

Moreover, taking the result of Zhoukou station as an example, the trivariate M3 copula joint cumulative probability distribution of $D–S–P$ is displayed in Figure 7. Given any two variables of $D,S$, and $P$, the joint probability would be increasing with the other variable’s increase, to a final fixed value. As shown in Figure 7, $D=5$ and $S=5$, the joint probability increases from 0 to 0.5; $D,S,$ and $P$ all grew without bound, the joint probability increased up to 1.

4.4. Drought Frequency Analysis

Evaluating droughts based on univariate frequency analysis may lead to over/underestimation of drought frequencies due to the high correction between drought variables. Consequently, multivariate frequency analysis is needed to estimate the joint return periods and appropriately evaluate the risk associated with drought events, as well as to plan and manage water use under drought conditions.

Given that the univariate return periods (T) were 2, 5, 10, 20, and 50 years, the $D, S,$ and $P$ were estimated by each univariate marginal distribution function, and then $T_{and}$ and $T_{or}$ were estimated by the bivariate and trivariate joint copula function with the $D, S,$ and $P$ which encompass the univariate return periods given above T, the results were presented in

Table 7. Return period of copula function of drought characteristic variables.

Hydrometric Station	T	D	S	P	D–S		D–P		S–P		D–S–P
					$T_{and}$	$T_{or}$	$T_{and}$	$T_{or}$	$T_{and}$	$T_{or}$	$T_{and}$	$T_{or}$
Luohe	2	2.40	2.54	1.20	1.87	2.15	1.54	2.84	1.83	2.21	1.52	2.33
	5	6.30	5.55	1.59	4.27	6.03	3.38	9.57	3.71	7.66	2.92	6.99
	10	9.25	8.20	1.81	7.63	14.52	6.52	21.42	6.42	22.62	4.86	15.27
	20	12.20	11.27	2.01	13.43	39.18	12.82	45.51	11.55	74.32	8.40	33.33
	50	16.10	16.06	2.25	29.21	173.55	31.72	118.04	26.65	402.84	18.57	94.67
	100	19.05	20.33	2.42	54.57	596.86	63.22	239.12	51.69	1530.84	35.31	206.93
Zhoukou	2	2.03	1.96	1.12	1.91	2.10	1.75	2.33	1.80	2.25	1.67	2.27
	5	5.29	6.53	1.73	4.45	5.70	3.69	7.75	3.92	6.89	3.38	6.89
	10	7.46	9.80	2.10	8.09	13.10	6.50	21.67	6.91	18.11	5.61	17.83
	20	9.53	12.99	2.46	14.27	33.40	11.73	67.74	12.30	53.53	9.36	49.31
	50	12.19	17.12	2.92	30.54	137.82	26.92	351.18	27.63	262.62	19.97	251.50
	100	14.16	20.49	3.26	56.17	455.40	51.99	1305.76	52.77	952.37	37.00	960.36

Notes: unit of T, $T_{and}$, and $T_{or}$ is year, unit of $D$ is month.

. The comparisons among joint bivariate, trivariate period ($T_{and}$ and $T_{or}$), and univariate return periods (T) showed that univariate return periods were always lower than $T_{and}$ and greater than $T_{or}$; and bivariate return periods ($T_{and}$ and $T_{or}$) were always greater than trivariate return periods ($T_{and}$ and $T_{or}$), and the joint return periods $T_{and}$ and $T_{or}$ were regarded as the upper and lower bounds of the return periods (T) from the univariate frequency analysis. For instance, in Luohe station, for the univariate return period $T=10$ year, the duration, severity, and intensity peak estimated by univariate marginal distribution were 9.25 months, 8.2, and 1.81, respectively. Furthermore, the bivariate joint return period $T_{and}$ ($D\ge 9.25\cup S\ge 8.25, D\ge 9.25\cup P\ge 1.81, S\ge 8.25\cup P\ge 1.81$) of any two variables determined by bivariate copula function were 7.63, 6.52, and 6.42, which was less than T; the $T_{or}$ ($D\ge 9.25\cap S\ge 8.25, D\ge 9.25\cap P\ge 1.81, and S\ge 8.25\cap P\ge 1.81$) were 14.52, 21.42, and 22.62, which is greater than T. The trivariate joint return period $T_{and}$ ($D\ge 9.25\cup S\ge 8.25\cup P\ge 1.81$) and $T_{or}$ ($D\ge 9.25\cap S\ge 8.25\cap P\ge 1.81$) were 4.86 and 15.27, which are 29% and 21% less than the average bivariate joint return period $\left(T_{and} \mathrm{and} T_{or}\right)$, respectively, and Tand of trivariate joint copula is 51% larger and $T_{or}$ is 57% less than the return period (T) when duration, severity, and intensity peak were considered independently by the univariate analysis.

Trivariate copula functions of D–P–S with the lower joint return periods ($T_{and}$ and $T_{or}$) used to designed frequency calculation could improve the security of the designed projects but always with more cost. Meanwhile, for the prediction and the research of complex drought-causing mechanisms identification, more attention was paid to the multivariate copula function with multiple drought characteristic variables, and the difficulty and uncertainty of the parameters estimation and function selection brought by the increasing the dimensions will be strengthened.

Given the $D, S,$ and $P$, the bivariate return period ($T_{and}$) of $D–S$ was the most, and the return period ($T_{or}$) of $D–S$ was the least among all the bivariate return periods, which indicated that there was the highest dependency between $D$ and $S$. Conversely, $T_{and}$ of $D–P$ were the least and the $T_{or}$ were the most, which indicated that the dependency between $D$ and $P$ was the lowest. The higher dependency between two drought variables, the higher probability of the other variable appeared when one variable appeared, which means the drought characteristic variables had higher dependency, and thus, the higher risk would be brought by the drought events with both higher variables. The correlation coefficients τ among the variables presented in Table 3 also support these options.

Figure 8 shows that the bivariate return periods ($T_{and}$) of all the bivariate and trivariate copula function in Zhoukou station were more than those in Luohe station, and the return period ($T_{or}$) in Zhoukou station was less than that in Luohe station which means that the drought characteristic variables in Zhoukou station were of higher dependency than that in Luohe station, and the greater drought risk would be brought by the drought events with the longer duration, the higher severity, and the higher intensity peak in the Luohe Station.

The results from the bivariate joint and trivariate return periods of droughts are expected to offer valuable information regarding development and management of water resources in the basin, the use of which will allow the effectiveness of various drought and water managements to be evaluated for different combinations of drought situations.

5. Conclusions

Frequency analysis of hydrological drought is important for agricultural areas. In this paper, a comprehensive assessment of drought frequency in SYRB was carried out using bivariate and trivariate frequency analysis that considers the inherent correlation among drought severity, duration, and intensity peak. SRI-3 time series and run theory derived based on runoff data from 1956–2008 were used to identify the drought events, and also characterize the different combination of drought severities, durations, and intensity peak. The main conclusions drawn from the study are as follows.

Based on the drought identification criterion of $SPI-3<-0.5$, there were 51 and 41 drought events with the average duration of 4.78 and 4.17 months, severity of 4.79 and 5.09, and the drought intensity peak of 1.35 and 1.50 in upstream basin (Zhoukou station) and downstream basin (Luohe station), respectively. Spatial distribution of droughts in the SYRB showed that the drought events in the upstream had greater frequency with a longer duration but lower severity and intensity peak than that in the downstream. This means that drought deficit in the downstream was more serious than that in the upstream.

The bivariate copulas of Frank, Clayton, Gumbel, and Joe were used to fit the bivariate joint distribution of hydrological drought variables. For univariate marginal distribution functions, the exponential and gamma distribution of the drought duration, log-normal, and Weibull distribution for the drought severity, and gamma and log-normal distribution of the intensity peak in upstream and downstream were determined, respectively. Further, in Luohe station, Frank copula was proven to be the best-fit bivariate joint distribution model of drought duration and severity; Gumbel copula was the best-fit distribution model for drought duration and intensity peak; and Clayton copula was fitted best for drought severity and intensity peak. In Zhoukou station, Frank copula was found to fit best for any two drought characteristic variables.

The trivariate joint distribution analysis of drought duration, severity, and intensity peak showed that Frank copula was considered the best-fitting trivariate joint distribution function of D, S, and P in Luohe station, and M3 copula fitted best of $D, S,$ and $P$ in Zhoukou station.

With the bivariate and trivariate copula functions, the joint return periods ($T_{and}$ and $T_{or}$) of all the drought variables were estimated. Comparisons analysis of univariate, bivariate, and trivariate return period showed that univariate return periods are always lower than $T_{and}$ and greater than $T_{or}$; and bivariate return periods ($T_{and}$ and $T_{or}$) are always greater than trivariate return periods ($T_{and}$ and $T_{or}$), and the joint return periods Tand and Tor are regarded as the upper and lower bounds of the return periods (T) from the univariate frequency analysis, which could improve the security of the designed projects but always with more cost.

The results from Luohe Station and Zhoukou station showed that the higher $T_{and}$ and lower $T_{or}$ of all the bivariate and trivariate copula function in Zhoukou station were more than those in Luohe station, which means that the drought characteristic variables were higher dependency, which showed the higher risk by the severe drought with higher severity, longer duration, and higher intensity peak.

Finally, to the best of our knowledge, the presented study is the first attempt to characterize the droughts in Shaying River Basin using joint frequency analysis by explicitly considering the correlation between drought severity, duration, and intensity peak. Due to the observed strong correlation, we believe the multivariate frequency analysis has improved drought assessment.

Since there are so many uncertainties in hydrological drought analysis proposed by some papers, including the drought events and character identification, the frequency distribution, parameter estimation, etc., many further studies on hydrological drought are still worth doing. For example, the uncertainty of the drought threshold defined to identify the droughts based on run theory ($SRI\le -0.5$ in this paper) which diminish or increase the number of droughts is still worth exploring. The study will further provide a framework for regarding climate trends and climate change future projection on seasonal time step, and offer valuable information for the development and management of water resources in the basin. Furthermore, this study will help to better understand the droughts, to reveal droughts occurrence laws, to reduce drought disasters, and to serve social practices.